Erlanger Programm: Talking Tonality

Nota Bene: all of this project (and consequently, of this article) is done with a tempered tuning/enharmonic shift mindset. In other words, e.g. C# = Db = Ebbb.

Early on in the creation of this project, I had thought about tonality – almost before I thought about anything else. Among the first ideas I had were the following:

  1. use of ecclesiastic modes (other than Ionian and Aeolian),
  2. use of non-heptatonic scales (i.e. more or less than seven notes per octave),
  3. a funny relationship between the roots/scales used (whatever “funny” means).

A Funny Relationship

So what does “a funny relationship” mean in the context of tonality? As always, I’m a lot about symmetry, and if we have four main parts, the most obvious choice would be to space their roots evenly over the twelve semitones, in other words every three semitones. As it so happens, this would mean that the roots form a diminished 7th chord, and there’s only three options: one including C and going C-Eb-Gb-A, the other one step down (or up) being B-D-F-Ab and C#-E-G-A# – in every case, starting from whichever note you wanted. I chose the second one, mainly because I like D and Ab a lot.

Talking about symmetry relationships, here we have our first one: if we would consider this set of roots a scale (with four notes), it would have a one-step symmetry (i.e. it looks the same in both directions and from every root note) with a step size of three semitones (minor 3rd). It just so happens that there’s only a total of three of those four-tone-scales, which together encompass all of the twelve semitones, but do not share a tone with each other. Great! Vector space!

Now a discussion of very theoretical/abstract stuff could ensue, which I initially wanted to put into an annex, but then I decided to spare you of. Was that grammar sound?

So essentially, that was step one: use four different tonalities, rooted on B, D, F and Ab, respectively.

Ecclesiastic Modes

I guess the word’s really “ecclesiastical”, no? If so, I’ve been using it wrongly for years, and will continue to do so. Kirchentonarten, for christ’s sake!

There’s seven of them, and all of them are essentially the same, only starting from a different note. You can imagine them as playing on a piano keyboard, using only white keys, and starting from different notes. The best-known ones are of course Ionian, aka major, starting from C, and Aeolian, aka natural minor, starting from A. The remaining ones are two more each with the root triad being minor (Dorian from D and Phrygian from E) or major (Lydian from F and Mixolydian from G), and the special case of Locrian (starting from B), where this triad is a diminished chord.

Of course, you can transpose those as well, so an Ab Lydian would be very well possible (and would then have three flats).

My idea here was to use one that was “more minor”, and one that was “more major”. I’ve always enjoyed Dorian a lot, and so that was the “more minor” choice. For the other, as Ionian would be too obvious, there was the choice between Lydian and Mixolydian. Now Lydian has a special trick up its sleeve, namely the fourth is an augmented one (read: tritone). Plus, it has a major dominant diatonically, which is not the case for Mixolydian – so I opted for Lydian.

Going Non-Heptatonic

Of course, the best-known non-heptatonic mode we know is the Blues pentatonic. I didn’t want to use that.

Of course, with me always raving about symmetry relations, an obvious choice would be one that has equidistant steps all over. Theory has it there can be one of those modes for every divisor of twelve, some of which have only theoretical value. And inversely, twelve divided by those divisors lead then to an uniform step size (in semitones). Furthermore, logic has it that the number of different scales available is then identical to that step size. Those modes are:

  • 12/12=1: chromatic (there’s only one, obviously),
  • 12/2=6: wholetone (with two of them),
  • 12/3=4: minor thirds (three),
  • 12/4=3: major thirds (four),
  • 12/6=2: tritones (six),
  • 12/1=1: octaves (twelve).

No need to explain the chromatic one: total freedom here, but for that reason, not that interesting if you don’t want to have that and only that. The “octaves” one is a rather degenerate choice (it only has one note: its root), and both the tritone one and the thirds one (which are cool if you only run around in either minor or major thirds) were things I didn’t want to use. So that left the wholetone one.

Whole-Tone

There’s quite some use of those in classical music – starting from Mozart’s Ein Musikalischer Spaß (although more meant as a joke or parody) up to widespread use in the work of Debussy, but also important appearances e.g. with Berg and Bartok (both heroes of mine).

The first trick with this scale with it’s what I’d like to call 1-symmetry (meaning it repeats every step identically) is that every possible chord is identical on every note.

Excursion: Naming Intervals in Non-Heptatonic Tonality

We usually name intervals based on the number of semitones, and if we’re in a rather diatonic, heptatonic mindset and with the exception of the tritone (which could be a diminished fifth or an augmented fourth), we’re safe and unambiguous. That means if we see four semitones, it’s always a major third, one semitone a minor second etc.

However, as we’re moving away from heptatonic tonality, I propose to call intervals based on the number of diatonic steps it takes.

Staying with a practical example: wholetones and starting on C:

The interval C-D is a second (in standard terms, a major one). The interval C-E is a (major) third. Now C-F# is a fourth (an augmented one), and C-G# is a fifth (augmented), not a minor sixth. We end with the the sixth of C-A# (augmented) and the seventh of C to C (an augmented one). This is important to keep in mind, also when we discuss other tricky tonalities later.

So coming back to the fact that the chords/interval are identical from every note. I already named them, in immediately we see two interesting properties:

  • the third is always major,
  • there is no such thing as a seven-semitone “perfect fifth”.

Can I do something like faking a II-V-I? I hope I don’t need to find out. Rather, let’s talk about the second option:

The “2-1”

Something nearly as simple as wholetone or chromatic: use wholetone steps and semitone steps alternatingly. There’s two possible options, namely starting either with a wholetone or a semitone step. While the second option has advantages (mainly that you can have both a third that’s a minor third and a fourth that’s a diminished fourth and looks like a major third), I opted for the first one for reasons which become apparent later in this text.

Again taking the example of a scale starting with C, we would then have:

C D Eb F F# G# A B

Sticking with the rule we established before for naming intervals, we have a major second, a minor third, a perfect fourth, a diminished fifth, minor sixth, diminished seventh and a diminished octave. All in all, a tonality which makes it impossible to construct anything close to standard harmony.

Talking about symmetry, here we have (in my wording) a 2-symmetry, meaning that starting from D, you have a rather different picture: minor second, minor third, diminished fourth (looking like a major third), diminished fifth, diminished sixth (looking like a fifth) and seventh, and double-diminished octave. The fun thing here is that we can build a sus4/6 that looks like a major chord. Or a sus6 that looks like a minor chord. But we can only use it on the even-numbered notes (i.e. not on the tonic or dominant).

Putting it together

Now looking at the chosen modes (Lydian, Dorian, wholetone and 2-1) and roots (D, F, Ab and B), I did some drawings. And thought about which tonality would benefit which piece (mainly taking into account playability on bass guitar). I ended up with this:

  • Part A: Ab Lydian
  • Part B: F 2-1
  • Part C: B Wholetone
  • Part D: D Dorian

Now I want to bring your attention back to another diagram:

EP_Scales

This diagram shows where, in the concept described above, where the notes of each piece are, and where they are not.

And there’s one important finding

There’s everything except for F#. F# is the forbidden note of Erlanger Programm!

And that really formed the end of the tonality considerations for me: I would, if I did decide to go all-crazy/chromatic later on, avoid F# (see devil/holy water analogy).

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s